Existence

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1. #1

   (ii) Starting from a different representative :

   $$\mathcal{E}'(h)=\sum W_t(i,j)\,d_G(g'^h(i,j),e)^2 = \sum W_t(i,j)\,d_G(g^{h_0\cdot h}(i,j),e)^2 = \mathcal{E}(h_0\cdot h).$$
2. #2

   Proof. Parametrise and . The flattening energy is:

   $$\mathcal{E}(\varphi) = \sum_{\substack{i,j\in F^\circ\\W_t(i,j)>0}} W_t(i,j)\;\bigl(1-\cos(\alpha_{ij}+\varphi_i-\varphi_j)\bigr).$$
3. #3

   Proof. Parametrise and . The flattening energy is: Step 1 (Exact convexity on the torus). Define . The energy is:

   $$\mathcal{E}(\varphi) = \sum_{(i,j)}W_t(i,j)\,(1-\cos\theta_{ij}).$$
4. #4

   Step 2 (Characterisation of critical points). At any critical point, for all :

   $$\sum_{j\sim i}W_t(i,j)\,\sin(\alpha_{ij}+\varphi_i-\varphi_j) = 0\quad\forall\,i\in F^\circ.$$
5. #5

   Step 4 (Hessian analysis and uniqueness). With all established, the Hessian of at has entries:

   $$H_{ii} = \sum_{j\sim i}W_t(i,j)\cos\theta_{ij}^*,\quad H_{ij}=-W_t(i,j)\cos\theta_{ij}^*\quad(i\ne j).$$
6. #6

   For : The optimal gauge solves a graph Laplacian system (Theorem 7.11). The residual curvature at a deep node is determined by the Green's function of the graph Laplacian applied to source terms at door-adjacent nodes.

   $$\rho_{F^\circ}(i)\;\le\;C\,e^{-\beta\,d_{\mathrm{graph}}(i,\Sigma)}\cdot\sum_{p\in\Sigma}e_p$$
7. #7

   Quantitatively:

   $$\sum_{i\in F_{\mathrm{deep}}}\rho(i)\;\le\;\alpha\cdot\mathcal{E}_{F^\circ}(h^*)\quad(\alpha<1),$$
8. #8

   Step 4 (Lojasiewicz convergence). By the Lojasiewicz gradient inequality (Fact B.8) applied to the real-analytic function on the compact manifold : there exist and such that near any critical value :

   $$\|\mathrm{grad}\,\mathcal{E}(h)\|\ge C\,|\mathcal{E}(h)-c|^\alpha.$$